[EM] Thoughts on Burial

[Juho]

On Jul 24, 2010, at 3:46 AM, fsimmons at pcc.edu wrote:

> You guys have come up with some interesting ideas about the  
> likelihood of sincere cycles, but my idea
> is not that complicated:
>
> Usually in the high stakes elections that I have witnessed there are  
> just a few issues that most voters
> feel strongly about, and opinions on these issues are highly  
> correlated (or anti-correlated) so that the
> voter distribution in issue space is basically cigar shaped.
>
> Perpendicular to the long axis of that cigar find a plane that  
> divides the voters into two equal subsets
> (plus or minus one).  The candidate closest to that plane is very  
> likely a Condorcet candidate.
>
> But this Condorcet candidate can be buried as easily as a Condorcet  
> candidate can be buried in a
> precisely one dimensional issue space.
>
> I like Condorcet methods that discourage burial in one dimensional  
> cases.  I don't care so much about
> the case where the candidates are distributed on the vertices of an  
> acute triangle, i.e. the triangle is
> close to equilateral.  In that case burial may serve a useful  
> purpose of decreasing the probability of
> winning for a low utility Condorcet candidate.
>
> In particular, the sincere profile
>
> 40 A>C>>B
> 30 B>C>A
> 30 C>A>>B
>
> could easily come from a one dim or cigar shaped issue space.  Any  
> condorcet method that doesn't
> make burial of C risky for the A faction in this context is going to  
> end up with more artificial cycles than
> real ones.

Thanks, this is at least a well defined case where strategic cycles  
might occur. (I guess "B>C>A" should be read "either B>C>A" or  
"B>>C>A", and in addition to "C>A>>B" votes there could be also some  
"C>A>B" and even few "C>B>A" votes.)

I'm not sure if this case would lead to artificial cycles very easily.  
75% of the A supporters should vote strategically to make the strategy  
work. A smaller number of strategic voters (50%) is sufficient to  
create an artificial cycle. There are many possible ways this strategy  
can fail. For example the B supporters prefer C to A. If they know  
that A supporters will try a strategy and win, then the B supporters  
might vote directly for C and thereby guarantee that the strategy of  
the A supporters will not work. The preferences may also change before  
the election, and part of the C supporters may sincerely rank A lower  
because of the attempted strategy that tries to steal the victory from  
their favourite. In short, if some society is so strategic that they  
would try this strategy then there could be also other strategic  
moves, and the whole election (and future elections) might become a  
chaos. It is possible that in some "very strategically oriented"  
societies with very stable opinions (e.g. assuming that B can not win  
even if A supporters would rank B higher, and C supporters will not  
stop liking A because of the strategy) we would get strategic cycles  
this way, but it seems probable to me that in most societies this kind  
of chaos would not emerge (assuming that some percentage of voters  
want to vote sincerely rather than steal the victory etc.) (sorry, no  
clear proof available).

>
> Note that random ballot on Smith is adequate for preventing the  
> burial without any defensive strategy on
> the part of the C supporters.
>
> On the other hand the profile
>
> 40 A>>C>B
> 30 B>>C>A
> 30 C>>A>B
>
> could not arise from a one dimensional or cigar shaped issue space.   
> And candidate C has such low
> uility, it wouldn't be bad if A got a share of the probability  
> through a burial of C.

One could also consider low utility Condorcet winners to be worth  
being elected with 100% probability. The difference in philosophy is  
if one tries to find a winner that would offer best sum of utility to  
the voters (=> sum of ratings like philosophy) or if one wants to find  
a winner that can rule the society thanks to having majority support.

Juho


>
> Random Ballot Smith doesn't discourage burial in this case, in which  
> C retains only 30% of the
> probability. Without more detailed information it would be  
> impossible to prove that C deserved more than
> that amount.
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